
On the right hand side of Eq. (REF ), the first exponential already resembles the path probability of a normal diffusive process, with a correction in the prefactor of the integral. This correction, which can be interpreted as a renormalization of the diffusion constant, can be set to zero by choosing \(\tau =e^{-\zeta }\)  (recall that \(\tau \)  is a free parameter). In this way the first exponential becomes equivalent to the Brownian case. The second exponential in Eq. (REF ), however,
reflects the non-Markovian character of the noise correlation function, as seen from the double time integral in the exponent (and the prefactor \(\epsilon \)  of the integral).
In this sense, the second exponential represents the signature of non-Markovianity within our first-order expansion.
We note that the result Eq. (REF ) matches the perturbative path probability of the FBM calculated in Ref. [1]}, [2]}, [3]}, [4]}, [5]}.
